# Loyola College B.Sc. Statistics April 2007 Distribution Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.  DEGREE EXAMINATION – STATISTICS

 AC 15

FOURTH SEMESTER – APRIL 2007

ST 4501  – DISTRIBUTION THEORY

Date & Time: 24/04/2007 / 9:00 – 12:00         Dept. No.                                                     Max. : 100 Marks

Section A

1. State the properties of a distribution function.
2. Define conditional  and marginal distributions.
3. What is meant by ‘Pair-wise Independence’ for a set of ‘n’ random variables.
4. Define Conditional Variance of a random variable  X given a  r.v. Y=y.
5. Define Moment Generating Function(M.G.F.) of a  random variable X.
6. Examine the validity of the statement “ X is a Binomial variate with mean 10 and standard deviation 4”.
7. If X is binomially distributed with parameters n and p, what is the distribution of

Y=(n-x)?

1. Define ‘Order Statistics’ and give an example.
2. State any two properties of Bivariate Normal distribution.
3.  State central limit theorem.

Section B

Answer any FIVE questions                                                                                                     (5×8 =40)

1. Let (X,Y) have the joint p.d.f. described as follows:

(X,Y) : (1,1)   (1,2)   (1,3)   (2,1)   (2,2)   (2,3)

f(X,Y) :  2/15   4/15    3/15    1/15   1/15   4/15. Examine if X and Y are independent.

1. The joint p.d.f. of  X1 and X2 is : f(x1, x2) =.

[a] Find the conditional p.d.f. of X1 given X2=x2.

[b] conditional mean and variance of X1 given X2=x2

1. Derive the recurrence relation for the probabilities of Poisson distribution
2. Obtain Mode of Binomial distribution
3. Obtain Mean deviation about mean of  Laplace distribution
4. The random variable X follows Uniform distribution over the interval(0,1). Find the

distribution of Y = -2 log X.

1. Obtain raw moments of Student’s ‘t’ distribution, Hence fine the Mean and the Variance.
2. Let Y1, Y2, Y3, and Y4 denote the order statistics of a random sample of size 4 from a

distribution having p.d.f.   f(x) =.       Find P (Y3 > ½ ).

Section C

Answer any TWO questions only                                                                                 (2×20 =40)

19.[a]  The random variables X and Y have the joint p.d.f. f(x, y) =.

Compute Correlation co-efficient between X and Y.                                                       [15]

[b] Let X and Y are two r.v.s with the p.d.f. f(x, y) =.

Examine whether X and Y are stochastically independent.                                            [5]

20 [a] Derive M.G.F. of Binomial distribution and hence find its mean and variance.

[b] Show that  E(Y| X=x) = (n-x) p2/ (1-p1),if (X,Y) has a Trinomial distribution with

parameters n, p1 and p2.

1. [a] Prove that Poisson distribution is a limiting case of Binomial distribution, stating the

assumptions involved.

[b] Let X and Y follow Bivariate Normal distribution with;m1=3  m2=1  s12 =16  s22 =25 and

r = 3/5.

Determine the following probabilities: (i) P[(3<Y<8)|X=7]   (ii) P[ (-3<X<3) | Y=4]

1. [a[ If X and Y are two independent Gamma variates with parameters m and n respectively.

Let U=X+Y and V= X / (X+Y)

[i] Find the joint p.d.f. of  U and V

[ii] Find the Marginal p.d.f.s’ of U and V

[iii] Show that the variables U and V are independent.                                                [12]

[b] Derive the p.d.f of F-variate with (n1, n2) d.f.                                                            [8]

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